Lecture Notes in Computer Science 3472

250 Verena Wolf
sP
(τ, 1
1
) (c, 3 3 )
(b, 1
3 )
u1
⊑ ste
CH
P ⊑ wte
CH
u2
u3
v1
w1
sQ
(τ, 1
1
) (τ, 2 2 )
v2
(c, 1) (b, 1)
w2
Fig. 9.6. P ∼ tr CH Q but P �∼ wte
CH Q.
compares each single trace probability, it is clear that P ⊑ste
CH
Q. Hence, for P, Q ∈ FPP we have
P ⊑ste CH Q =⇒ P ⊑wte CH Q =⇒ P ⊑tr CH Q.
z1
b
sT
c
z2
Q implies
In the following we will see that these implications can not be reversed. The
superscript is chosen due to the fact that ⊑wte CH is a ”weak testing preorder” com-
(”strong testing preorder”). We denote the in-
pared to the finer preorder ⊑ ste
CH
duced equivalence relations by ∼ i CH = ⊑i CH ∩ (⊑i CH )−1 where i ∈{tr, wte, ste}
and sometimes we write ⊑CH and ∼CH instead of ⊑ ste
CH
and ∼ste
CH
, respectively.
Example. Figure 9.6 shows two fully probabilistic processes P and Q with P ∼ tr CH
Q but P �∼ wte
CH
�
a∈Act
Q. The latter can be seen by applying the test process T .Then
�
trace 2
PrP�T (a) = 3 < a∈Act
If we apply some T ′ ∈T np,re
seq
for all α ∈ Act ∗ .
trace PrQ�T (a) =1.
instead we always have Pr trace
P�T ′(α) =Pr trace
Q�T ′(α)
In Figure 9.7 we have P ′ ∼ wte
CH Q ′ but P ′ �∼ ste
CH Q ′ . The latter can be seen if we
apply the test process T in Figure 9.6 to P ′ and Q ′ .Wehavethat
1
4
= Pr trace
P ′ �T
trace
(c) < PrQ ′ 1
�T (c) = 3 .
It is easy to see that P ′′ ∼ ste
CH Q ′′ .
⊓⊔
Note that we obtain the same relations in Definition 9.14 by replacing the
probability distribution Pr trace
P (·) by the conditional probability distribution
Pr ctrace
P (·) defined as follows: For α ∈ Act ∗ and a ∈ Act let
Pr ctrace
�
trace PrP (α a)/Pr
P (α a) =
trace
P (α) if Pr trace
P (α) > 0,
0 otherwise.
This is an important fact, on which the proof of the characterization theorem 9.29
relies.